Statistical Genetics

• Associacy Matrices in Two-Associate Class PBIBD:
• In a two-associate PBIBD, parameters satisfy the identity relations:

• Given $n_1 = 6$ and $n_2 = 2$:
• For $P^1$ (first associates):
• Row 1: $p_{11}^1 + p_{12}^1 = 6 - 1 = 5 \implies 3 + 2 = 5$
• Row 2: $p_{21}^1 + p_{22}^1 = n_2 = 2 \implies 2 + 0 = 2$
• Hence, $P^1 = \begin{pmatrix} 3 & 2 \\ 2 & 0 \end{pmatrix}$
• For $P^2$ (second associates):
• Row 1: $p_{11}^2 + p_{12}^2 = n_1 = 6 \implies 6 + 0 = 6$
• Row 2: $p_{21}^2 + p_{22}^2 = n_2 - 1 = 2 - 1 = 1 \implies 0 + 1 = 1$
• Hence, $P^2 = \begin{pmatrix} 6 & 0 \\ 0 & 1 \end{pmatrix}$
• Conclusion: Matches Option 2.

Correct option: (2) ana p?= (5 ° 21 ol 2 as 4 ana p?= (¢ e 20 o1

• Balanced Lattice Design Construction and Properties:
• Lattice Design is an incomplete block design where the number of treatments $v$ is a perfect square ($v = s^2$). For $s = 3$, $v = 3^2 = 9$ treatments in blocks of size $k = s = 3$.
• Statement (A): By utilizing the complete set of $s - 1 = 2$ Mutually Orthogonal Latin Squares (MOLS), a complete set of $s + 1 = 4$ replicates is constructed, ensuring that every pair of treatments occurs together in the same block exactly $\lambda = 1$ time (Balanced Lattice). (True)
• Statement (B): The parameters of this balanced lattice are $v = 9, b = 12, r = 4, k = 3, \lambda = 1$, which satisfies all BIBD conditions: $vr = bk = 36$ and $\lambda(v - 1) = r(k - 1) = 8$. (True)
• Statement (C): Blocks X, XI, and XII are generated by superimposing Latin squares on the standard array. (True)
• Statement (D): The number of treatments $v = 9$ is a square and block size $k = 3 = \sqrt{v}$. (True)
• Conclusion: (A), (C) and (D) only (Option 3).

Correct option: (3) (A), (C) and (D) only.

• Identifying Confounded Factorial Interactions from Layout:
• In a $2^4$ factorial experiment with 2 blocks per replication, the interaction confounded corresponds to the defining contrast linear combination $L = \sum \alpha_i x_i \pmod 2$.
• Replication I Analysis:
• Check Block 1 treatments: 0000, 0011, 0101, 0110, 1001, 1010, 1100, 1111.
• Sum of all four coordinates: $L = x_1 + x_2 + x_3 + x_4 \pmod 2$.
• $0+0+0+0 = 0$
• $0+0+1+1 = 0$
• $0+1+0+1 = 0$
• $0+1+1+0 = 0$
• $1+0+0+1 = 0$
• $1+0+1+0 = 0$
• $1+1+0+0 = 0$
• $1+1+1+1 = 0$
• All units in Block 1 yield $L \equiv 0 \pmod 2$, whereas units in Block 2 yield $L \equiv 1 \pmod 2$. Thus, ABCD is confounded in Replication I.
• Replication II Analysis:
• Check Block 1 treatments: 0000, 0001, 0110, 0111, 1010, 1011, 1100, 1101.
• Sum of first three coordinates: $L = x_1 + x_2 + x_3 \pmod 2$.
• All units in Block 1 yield $x_1 + x_2 + x_3 \equiv 0 \pmod 2$, whereas units in Block 2 yield $x_1 + x_2 + x_3 \equiv 1 \pmod 2$.
• Thus, ABC is confounded in Replication II.
• Conclusion: ABCD in Replication I and ABC in Replication II (Option 4).

Correct option: (4) ABCD in Replication | and ABC in Replication II